3. Potential density D is defined as the density of which dry air would attain if it were transformed reversibly and adiabatically from its existing conditions to a standard pressure p. (usually 1000 mb). If the density and pressure of the air are p and p, respectively, it can be proven that the potential density can be expressed as, D = P(P₁ Р ojev / cp where cp and c, are the specific heats of air at constant pressure and constant volume, respectively. (a). Calculate the potential density of a quantity of air at a pressure of 600 mb and temperature -15 °C. (b) Show that 1 dD D dz 1 T d -r) = where I'd is the dry adiabatic lapse rate, I the actual lapse rate of the atmosphere, and T the temperature at height z. [hint: Cp - Cv R where R is the gas constant for dry air (R=287 Jdeg¹ kg-¹), cp-1004 Jdeg¹ kg¹, and cv-717 Jdeg¹ kg¹). (c) Show that the criteria for stable, neutral, and unstable conditions in the atmosphere are that the potential density decreases with height, is constant with height, and increases with height, respectively. (Note: If Ia > I, stable; If Id = r, neutral; Id
3. Potential density D is defined as the density of which dry air would attain if it were transformed reversibly and adiabatically from its existing conditions to a standard pressure p. (usually 1000 mb). If the density and pressure of the air are p and p, respectively, it can be proven that the potential density can be expressed as, D = P(P₁ Р ojev / cp where cp and c, are the specific heats of air at constant pressure and constant volume, respectively. (a). Calculate the potential density of a quantity of air at a pressure of 600 mb and temperature -15 °C. (b) Show that 1 dD D dz 1 T d -r) = where I'd is the dry adiabatic lapse rate, I the actual lapse rate of the atmosphere, and T the temperature at height z. [hint: Cp - Cv R where R is the gas constant for dry air (R=287 Jdeg¹ kg-¹), cp-1004 Jdeg¹ kg¹, and cv-717 Jdeg¹ kg¹). (c) Show that the criteria for stable, neutral, and unstable conditions in the atmosphere are that the potential density decreases with height, is constant with height, and increases with height, respectively. (Note: If Ia > I, stable; If Id = r, neutral; Id
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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![**Potential Density in Atmospheric Science**
Potential density, \( D \), is defined as the density that dry air would achieve if it were transformed reversibly and adiabatically from its current conditions to a standard pressure, \( p_0 \) (commonly 1000 mb). If the density and pressure of the air are \( \rho \) and \( p \), respectively, potential density can be expressed as:
\[ D = \rho \left( \frac{p_0}{p} \right)^{c_v/c_p} \]
where \( c_p \) and \( c_v \) are the specific heats of air at constant pressure and constant volume, respectively.
**Problem Overview:**
(a) **Calculate the Potential Density**
- Given: a quantity of air at a pressure of 600 mb and a temperature of -15°C.
(b) **Derive the Rate of Change of Potential Density**
- Show that:
\[
\frac{1}{D} \frac{dD}{dz} = -\frac{1}{T} (\Gamma_d - \Gamma)
\]
where \( \Gamma_d \) is the dry adiabatic lapse rate, and \( \Gamma \) is the actual lapse rate of the atmosphere. \( T \) is the temperature at height \( z \).
*Hint*: \( c_p - c_v = R \), where \( R \) is the gas constant for dry air (\( R = 287 \, \text{J deg}^{-1} \text{kg}^{-1} \)), \( c_p = 1004 \, \text{J deg}^{-1} \text{kg}^{-1} \), and \( c_v = 717 \, \text{J deg}^{-1} \text{kg}^{-1} \).
(c) **Stability Analysis**
- Demonstrate that the criteria for stable, neutral, and unstable conditions in the atmosphere are that potential density decreases with height, is constant with height, and increases with height, respectively.
*Note*:
- If \( \Gamma_d > \Gamma \), the atmosphere is stable.
- If \( \Gamma_d = \Gamma \), the atmosphere is neutral.
- If \( \Gamma_d < \Gamma \), the atmosphere is unstable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ea503b5-9839-4a05-b482-50f8a035eb54%2Fe2b0409a-9627-473b-84b8-c993c89c024c%2Fen80mszb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Potential Density in Atmospheric Science**
Potential density, \( D \), is defined as the density that dry air would achieve if it were transformed reversibly and adiabatically from its current conditions to a standard pressure, \( p_0 \) (commonly 1000 mb). If the density and pressure of the air are \( \rho \) and \( p \), respectively, potential density can be expressed as:
\[ D = \rho \left( \frac{p_0}{p} \right)^{c_v/c_p} \]
where \( c_p \) and \( c_v \) are the specific heats of air at constant pressure and constant volume, respectively.
**Problem Overview:**
(a) **Calculate the Potential Density**
- Given: a quantity of air at a pressure of 600 mb and a temperature of -15°C.
(b) **Derive the Rate of Change of Potential Density**
- Show that:
\[
\frac{1}{D} \frac{dD}{dz} = -\frac{1}{T} (\Gamma_d - \Gamma)
\]
where \( \Gamma_d \) is the dry adiabatic lapse rate, and \( \Gamma \) is the actual lapse rate of the atmosphere. \( T \) is the temperature at height \( z \).
*Hint*: \( c_p - c_v = R \), where \( R \) is the gas constant for dry air (\( R = 287 \, \text{J deg}^{-1} \text{kg}^{-1} \)), \( c_p = 1004 \, \text{J deg}^{-1} \text{kg}^{-1} \), and \( c_v = 717 \, \text{J deg}^{-1} \text{kg}^{-1} \).
(c) **Stability Analysis**
- Demonstrate that the criteria for stable, neutral, and unstable conditions in the atmosphere are that potential density decreases with height, is constant with height, and increases with height, respectively.
*Note*:
- If \( \Gamma_d > \Gamma \), the atmosphere is stable.
- If \( \Gamma_d = \Gamma \), the atmosphere is neutral.
- If \( \Gamma_d < \Gamma \), the atmosphere is unstable.
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